Question

Statistics I – First Analysis Project

You are part of a research team that is studying academic motivation in high school students. A question has arisen regarding the strength of the relationship between academic motivation and achievement. Some members of the team believe there is a strong positive relationship between academic motivation and achievement, while others don’t think the two variables are related much, if at all.

To provide some information relevant to this discussion, a small study was designed. A random sample of 50 high school seniors from the local school district agreed to participate. Each student filled out a measure of academic motivation and released his/her SAT-math scores to the researchers. The data for this study are provided on the following page.

The measure of academic motivation was the Peabody Academic Motivation Scale (PAMS). Studies of the reliability and validity of scores obtained from using this instrument have been favorable when the instrument was used with high school students. The possible scores range from 0 to 100. The SAT-math scores can range from 200 to 800 and have also been shown to be reasonably reliable and valid in previous studies.

For this project you should:

1. Analyze each of the variables individually

2. Analyze the relationship between the two variables

3. Write a one or two paragraph summary of the results

Please turn in (1) the summary, (2) the program, and (3) the output.

OBS SAT    MOT

OBS   SAT    MOT

1     710     68                                 2     660     74                                3     430     88                                4     570     90                                5     430     70                                6     560     37                                7     670     79                                8     420     50                                  9     440     77                               10    320     34                               11    650     82                               12    630     55                               13    440     79                               14    420     60                               15    490     89                               16    530     53                               17    490     48                               18    450     55                               19    550     97                               20    490     46                               21    490     71                               22    470     29                               23    680     82                               24    310     68                               25    370     31

26    660    100                               27    430     41                               28    550     50                               29    630     63                               30    400     23                               31    460     46                               32    450     76                               33    230     28                               34    600     57                               35    340     85                               36    510     70                               37    620     96                               38    530     68                               39    490     81                               40    410     50                               41    480     67                               42    400     54                               43    610     54                               44    510    71                               45    380     44                               46    530     85                               47    720     76                               48    510     67                               49    520     63                               50    470     56

Answer 1

Descriptive Statistics: SAT, MOT

Variable   Mean StDev CoefVar Minimum     Q1      Median     Q3 Maximum Range
SAT       502.2 109.3    21.75         230.0      430.0   490.0     577.5    720.0    490.0
MOT       63.66 19.32    30.35        23.00      50.00   67.00     79.00   100.00   77.00

Variable    IQR Skewness Kurtosis
SAT       147.5      0.03     -0.23
MOT       29.00     -0.18     -0.66

From the above plots we can conclude that two data come from normal distributions.

From above scatter plot, we observe that two variables are positively correlated but the relation is not so good.

This fact is also observed from the value of correlation coefficient where

Pearson correlation of SAT and MOT = 0.452.

Regression Analysis: SAT versus MOT

The regression equation is
SAT = 339 + 2.56 MOT

Predictor    Coef SE Coef     T      P
Constant   339.42    48.38 7.02 0.000
MOT        2.5571   0.7278 3.51 0.001

S = 98.4483   R-Sq = 20.5%   R-Sq(adj) = 18.8%

Analysis of Variance

Source          DF      SS      MS      F      P
Regression       1 119639 119639 12.34 0.001
Residual Error 48 465219    9692
Total           49 584858

Since P-value of ANOVA test is less than 0.05 so the regression equation is significant but only 20.5% of total variation of SAT score is explained by MOT through this regression equation.

From normal probability plot, we observe that the residuals follow normal distribution. From residual plot we see that assumption of linearity holds since the points do not follow any definite pattern and assumption of equal variances also holds.